MathNet 2011 Question 57

Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $\widehat{BAC}$ of $\Omega$. Prove that lines $BD$, $CE$, $OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.

(This problem was suggested by Qiang Song.)